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          Problem 652
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<h1 id="Problem-652"><a href="#Problem-652" class="headerlink" title="Problem 652"></a><a href="https://projecteuler.net/problem=652" target="_blank" rel="noopener">Problem 652</a></h1><hr>
<h2 id="Distinct-values-of-a-proto-logarithmic-function"><a href="#Distinct-values-of-a-proto-logarithmic-function" class="headerlink" title="Distinct values of a proto-logarithmic function"></a><strong>Distinct values of a proto-logarithmic function</strong></h2><p>Consider the values of $\log_2(8)$, $\log_4(64)$ and $\log_3(27)$. All three are equal to $3$.</p>
<p>Generally, the function $f(m,n)=\log_m(n)$ over integers $m,n\ge 2$ has the property that<br>$f(m_1,n_1)=f(m_2,n_2)$ if</p>
<ol>
<li>$m_1=a^e$, $n_1=a^f$, $m_2=b^e$, $n_2=b^f$ for some integers $a, b, e, f$ or</li>
<li>$m_1=a^e$, $n_1=b^e$, $m_2=a^f$, $n_2=b^f$ for some integers $a, b, e, f$</li>
</ol>
<p>We call a function $g(m,n)$ over integers $m,n\ge 2$ <i>proto-logarithmic</i> if</p>
<ul>
<li>$g(m_1,n_1) = g(m_2,n_2)$ if any integers $a, b, e, f$ fulfilling 1. or 2. can be found</li>
<li><strong>and</strong> $g(m_1,n_1)\neq g(m_2,n_2)$ if no integers $a, b, e, f$ fulfilling 1. or 2. can be found</li>
</ul>
<p>Let $D(N)$ be the number of distinct values that any proto-logarithmic function $g(m,n)$ attains over $2\le m,n\le N$.<br>For example, $D(5)=13$, $D(10)=69$, $D(100)=9607$ and $D(10000)=99959605$.</p>
<p>Find $D(10^{18})$, and give the last $9$ digits as answer.</p>
<p><strong>Note</strong>: According to the <strong>four exponentials conjecture</strong> the function $\log_m(n)$ is <i>proto-logarithmic</i>. While this conjecture is yet unproven in general, $\log_m(n)$ can be used to calculate $D(N)$ for small values of $N$.</p>
<hr>
<h2 id="原型对数函数的不同取值"><a href="#原型对数函数的不同取值" class="headerlink" title="原型对数函数的不同取值"></a><strong>原型对数函数的不同取值</strong></h2><p>$\log_2(8)$、$\log_4(64)$和$\log_3(27)$这三个表达式都等于$3$。</p>
<p>一般地，定义在整数$m,n\ge 2$上的函数$f(m,n)=\log_m(n)$有如下性质：<br>$f(m_1,n_1)=f(m_2,n_2)$仅当</p>
<ol>
<li>$m_1=a^e$，$n_1=a^f$，$m_2=b^e$，$n_2=b^f$，其中$a, b, e, f$均为整数；或者</li>
<li>$m_1=a^e$，$n_1=b^e$，$m_2=a^f$，$n_2=b^f$，其中$a, b, e, f$均为整数。</li>
</ol>
<p>我们称定义在整数$m,n\ge 2$上的函数$g(m,n)$为<i class=zh>原型对数函数</i>，如果它满足</p>
<ul>
<li>如果存在满足1. 或2. 的整数组$a, b, e, f$，则$g(m_1,n_1) = g(m_2,n_2)$； </li>
<li><strong>并且</strong>，如果不存在满足1. 或2. 的整数组$a, b, e, f$，则$g(m_1,n_1) \neq g(m_2,n_2)$。</li>
</ul>
<p>记$D(N)$记为任意原型整数函数$g(m,n)$在$2\le m,n\le N$上不同取值的数目。<br>已知，$D(5)=13$，$D(10)=69$，$D(100)=9607$，$D(10000)=99959605$。</p>
<p>求$D(10^{18})$，并给出最后$9$位数字作为你的答案。</p>
<p><strong>注意</strong>：根据<strong>四指数猜想</strong>，函数$\log_m(n)$是<i class=zh>原型对数函数</i>。尽管这个猜想尚未被证实，但是对较小的$N$可以用$\log_m(n)$来计算$D(N)$。</p>
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          <div class="post-toc motion-element"><ol class="nav"><li class="nav-item nav-level-1"><a class="nav-link" href="#Problem-652"><span class="nav-text">Problem 652</span></a><ol class="nav-child"><li class="nav-item nav-level-2"><a class="nav-link" href="#Distinct-values-of-a-proto-logarithmic-function"><span class="nav-text">Distinct values of a proto-logarithmic function</span></a></li><li class="nav-item nav-level-2"><a class="nav-link" href="#原型对数函数的不同取值"><span class="nav-text">原型对数函数的不同取值</span></a></li></ol></li></ol></div>
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